Abstract: Self-adaptive numerical methods provide a powerful and automatic approach in scientific computing. In particular, Adaptive Mesh Refinement (AMR) algorithms have been widely used in computational science and engineering and have become a necessary tool in computer simulations of complex natural and engineering problems. The key ingredient for success of self-adaptive numerical methods is a posteriori error estimates that are able to accurately locate sources of global and local error in the current approximation. Talks in this mini-symposium will cover some recent advances in the development and analysis of both a posteriori estimators and (convergent) adaptive schemes, as well as indicate directions of future research.

MS-Mo-D-22-314:30--15:00A posteriori error estimation using auxiliary subspace techniquesOvall, Jeffrey (Portland State Univ.)Abstract: The discretization error for conforming simplicial finite elements is estimated by computing a function in an auxiliary subspace. The corresponding error estimates are proven to be efficient and reliable (up to an oscillation term), and numerical experiments demonstrate its robustness with respect to singularities, variation in the coefficients of the differential operator, and polynomial degree used in the discretization.

MS-Mo-D-22-415:00--15:30Optimality of adaptive finite element methods for controlling local energy errorsDemlow, Alan (Texas A&M Univ.)Abstract: While proof of convergence and optimality of adaptive FEM (AFEM) for controlling standard energy errors is now relatively standard, there are few corresponding results concerning optimality of AFEM for controlling other norms of the error. In this talk we discuss optimality of an AFEM for controlling local energy norms of the error.